Interpreting Logrank Statistic Trajectories

What a Logrank Trajectory Shows

At an event-driven interim analysis, the logrank statistic summarizes the treatment-control difference accumulated over the observed event history. A trajectory is the sequence of these cumulative statistics at successive data cutoffs. Reading the path well means separating three features: the direction of the score, the amount of information accumulated, and whether a prespecified sequential boundary has been crossed.

The trajectory is not a sequence of independent tests, nor is each change in the standardized statistic a direct estimate of how the treatment effect changed between looks. Each look reuses the event history from earlier looks. Consequently, the cumulative statistics are dependent, and their scale changes as information accumulates.

Definition: Use the convention that the cumulative logrank score \(U_k\) at look \(k\) is observed minus expected events in the treatment group, with expectation calculated under the null hypothesis and the logrank risk-set procedure. Let \(V_k\) be the corresponding cumulative variance, or information, and define the cumulative logrank statistic as \(Z_k=U_k/\sqrt{V_k}\), when \(V_k>0\). Under this convention, a negative score points toward fewer events in the treatment group and thus toward treatment benefit; a positive score points toward more events and thus toward harm. If a report uses the opposite score convention, all signs must be reversed when interpreting direction.

This sign convention describes the statistical direction, not a causal conclusion by itself. Interpretation also depends on the study design, outcome definition, and analysis plan. For a stratified logrank test, the score and variance are combined across the prespecified strata; the same sign interpretation applies when the treatment-group score uses this convention.

Reading Direction and Information Together

Information is nondecreasing across nested looks: \(V_k\geq V_{k-1}\). It is not necessarily proportional to the number of participants enrolled or even to the number of events. It reflects the variance of the score given the observed risk sets and allocation. The earlier material on realized information fractions establishes how to compare the information at a look with the planned information scale; here the focus is what that accumulation means for the path of \(Z_k\).

A negative \(Z_k\) says that the cumulative score favors fewer treatment-group events relative to the null expectation. Its magnitude measures the score relative to its standard error, \(\sqrt{V_k}\). A statistic can become less negative even while the cumulative score remains negative: later score increments may be less favorable, or information may increase faster than the magnitude of the score. Therefore, “the Z statistic moved toward zero” is not the same statement as “the cumulative score changed sign.”

A useful local description of the score is the score-based estimate \(U_k/V_k\), with approximate standard error \(1/\sqrt{V_k}\). In a proportional-hazards score formulation, this is a first-order estimate of the treatment log hazard ratio near the null, under the same sign convention. It is an approximation, not a replacement for the model’s fitted treatment-effect estimate. A more negative score-based estimate indicates a stronger estimated benefit direction; the standardized statistic also depends on information.

Why Cumulative Statistics Are Correlated

The dependence between looks follows directly from cumulative scoring. Write \(\Delta U_k=U_k-U_{k-1}\) and \(\Delta V_k=V_k-V_{k-1}\), with \(U_0=V_0=0\). Under the null and the usual large-sample score approximation, successive score increments have mean zero and variance approximately equal to their information increments. For nested looks, this gives the following covariance structure.

\operatorname{Cov}(U_i,U_j)\approx V_i \quad (i\leq j), \qquad \operatorname{Corr}(Z_i,Z_j)\approx\sqrt{\frac{V_i}{V_j}}.

The second relation shows why treating every look as a fresh, independent test is wrong: later cumulative statistics contain earlier score information. It also explains why sequential boundaries are calibrated for the full monitoring plan rather than chosen by applying a single-look cutoff independently at every look.

The cumulative statistic can be updated exactly from the score and information increments:

Z_k = \frac{U_{k-1}+\Delta U_k}{\sqrt{V_{k-1}+\Delta V_k}} = \sqrt{\frac{V_{k-1}}{V_k}}Z_{k-1} + \frac{\Delta U_k}{\sqrt{V_k}}.

This identity is a useful reading aid. The first term carries forward the prior cumulative result, rescaled to the new information; the second term adds the new score contribution. In particular, \(Z_k-Z_{k-1}\) is not the standardized score contribution from the new interval. The new interval’s score divided by its own standard deviation is \(\Delta U_k/\sqrt{\Delta V_k}\), when \(\Delta V_k>0\).

Worked Example: A Favorable Path That Levels Off

Suppose a hypothetical trial reports cumulative scores and information at three looks as follows. The score units are those of the logrank score, and information is in the corresponding score-variance units.

Look	\(U_k\)	\(V_k\)
1	-10	25
2	-20	64
3	-24	100

At each look, calculate \(Z_k=U_k/\sqrt{V_k}\). At look 1, \(Z_1=-10/\sqrt{25}=-2.00\). At look 2, \(Z_2=-20/\sqrt{64}=-2.50\). At look 3, \(Z_3=-24/\sqrt{100}=-2.40\). The cumulative score is negative throughout, so its direction favors fewer events in the treatment group. The standardized evidence becomes more negative by look 2, then moves slightly back toward zero by look 3.

The score-based estimates are \(U_k/V_k\): \(-10/25=-0.400\), \(-20/64=-0.3125\), and \(-24/100=-0.240\). These estimates are still in the benefit direction but move toward zero. The result does not establish that the treatment effect weakened over time: the quantities are cumulative, subject to sampling variation, and the score-based values are approximations. Whether any look meets a stopping criterion must be judged against the prespecified boundary, not from the shape alone.

Interpreting a Boundary Crossing

A boundary crossing is a design-based decision, not a visual description. For example, a two-sided design may specify a lower boundary \(B_k^-\) for benefit and an upper boundary \(B_k^+\) for harm at look \(k\). Under the sign convention above, benefit evidence crosses when \(Z_k\leq B_k^-\), while harm evidence crosses when \(Z_k\geq B_k^+\). The actual boundary values depend on the design, spending rule, information timing, and number of looks. The boundary application, not a generic cutoff, determines the sequential decision.

Worked Example: Distinguishing a Crossing From a Near Miss

Consider a hypothetical three-look design whose prespecified lower efficacy boundaries are \(-2.40\), \(-2.10\), and \(-2.00\). These values are illustrative only. Suppose the observed cumulative statistics are \(-2.20\), \(-2.15\), and \(-1.95\), respectively.

At look 1, \(-2.20>-2.40\), so the lower boundary is not crossed. At look 2, \(-2.15\leq-2.10\), so the lower boundary is crossed. The planned efficacy stopping criterion is met at look 2, assuming the design calls for stopping at that crossing and no other protocol condition changes the decision. The third statistic is not needed to reverse or confirm this decision; it would not be observed in the same way if the study stopped at look 2.

A reader should not substitute a remembered single-look critical value for these boundaries, or call the look-1 result a crossing because it is “close.” The criterion is the specified inequality at the actual analysis look.

If the monitoring plan includes futility rules, those are interpreted using their prespecified statistic, direction, and stopping convention as well. Efficacy and futility rules need not use the same inequality. A path that appears promising is not itself evidence that a futility rule was avoided, just as a path that appears weak does not authorize an unplanned stopping decision.

When the Path Reverses

Worked Example: A Later Increment Offsets Earlier Evidence

Suppose a hypothetical trial has \(U_1=-12\) and \(V_1=36\) at its first look, giving \(Z_1=-12/\sqrt{36}=-2.00\). At the next look, the cumulative values are \(U_2=8\) and \(V_2=100\), giving \(Z_2=8/\sqrt{100}=0.800\).

The later increments are \(\Delta U_2=8-(-12)=20\) and \(\Delta V_2=100-36=64\). The new interval’s standardized score contribution is \(20/\sqrt{64}=2.50\), which offsets the earlier negative score and leaves the cumulative score positive. The identity for the update gives \(Z_2=\sqrt{36/100}(-2.00)+20/\sqrt{100}=-1.20+2.00=0.800\), matching the direct calculation.

The path has changed direction in its cumulative score: the final score now points toward more treatment-group events under the stated convention. This does not, on its own, establish that the treatment caused harm or that its effect changed at a particular time. Random variation, the changing risk sets, and the accumulated event history all matter. A formal boundary comparison and the prespecified analysis remain necessary.

Diagnostics, Comparisons, and Limits

Before interpreting a path, check that each plotted value is the cumulative statistic for the intended look and that the sign convention is consistent across tables, software, and boundary definitions. Confirm that the information values are cumulative and nondecreasing, and that reported \(Z_k\) values agree, up to rounding, with \(U_k/\sqrt{V_k}\). A negative information increment, a mismatch between the score and standardized statistic, or a direction label inconsistent with the score definition calls for checking the calculation and data cutoff before drawing a conclusion.

For understanding a trajectory, it helps to compare three summaries rather than relying on \(Z_k\) alone:

Cumulative score \(U_k\): the net observed-minus-expected treatment event contribution to date, with direction given by the stated convention.
Information \(V_k\): the scale of precision accumulated to date. A large increase in information can change the standardized statistic even when the score changes little.
Standardized statistic \(Z_k\): the cumulative score measured in its estimated standard-error units, to be compared with the design’s boundary at that look.

A sequence of single-look tests using a fixed nominal significance level is not an appropriate alternative to group sequential boundary monitoring, because it does not in general control the planned overall type I error. Nor is simply comparing each \(Z_k\) with the final-look boundary appropriate: boundaries can differ by look. The sequential boundaries or spending-based decision rules specified for the design are the relevant comparison.

A trajectory is also not a substitute for a model-based effect estimate with an interval, a review of data quality, or a prespecified sensitivity analysis. In particular, bends or reversals in a cumulative logrank path are not, by themselves, a diagnostic test for proportional hazards or proof of a time-varying effect. Those conclusions require methods and assumptions beyond visual inspection of this path. Event timing, censoring, risk-set composition, and chance can all influence the observed increments.

The practical value of interpreting trajectories is that it makes an interim decision auditable: the analyst can state the direction of the cumulative score, how much information has accrued, how the statistic changed, and whether the relevant boundary was crossed. It also prevents over-reading a visually striking path as a direct record of the treatment effect at each moment in follow-up.

Common pitfall: “The Z statistic became less negative, so the cumulative treatment effect reversed.” This reasoning confuses a standardized cumulative statistic with an interval-specific effect. Check \(U_k\) to see whether the cumulative score actually changed sign, inspect \(\Delta U_k\) and \(\Delta V_k\) to understand the new contribution, and describe the path as evidence subject to variation. A causal or time-varying-effect conclusion requires additional analysis.

Check Your Understanding

Use the convention \(U_k=O_T-E_T\), so negative scores point toward fewer treatment-group events. Solutions follow the questions.

If \(U=-15\) and \(V=100\), calculate \(Z\) and state its direction.
At two nested looks, \(V_1=25\) and \(V_2=100\). Under the null approximation, what is the correlation between \(Z_1\) and \(Z_2\)?
A cumulative score changes from \(-8\) to \(-3\). Has the cumulative score reversed direction? What can be said about the later score increment?
At a look, \(Z=-2.05\) and the prespecified lower boundary is \(-2.10\). Is there a crossing at that look?
Why is it incorrect to treat the differences between successive cumulative \(Z\) statistics as independent interval-specific tests?

Solutions. (1) \(Z=-15/\sqrt{100}=-1.50\), which is in the benefit direction under the stated convention, but its decision significance depends on the design boundary. (2) The correlation is approximately \(\sqrt{25/100}=0.500\). (3) No. The score remains negative; the later increment is \(-3-(-8)=5\), a positive contribution that offsets some earlier negative score. (4) No, because \(-2.05>-2.10\); the statistic has not reached or passed the lower boundary. (5) Cumulative statistics share earlier score information and are correlated; their differences are not the new interval score standardized by its own information. The increment is \(\Delta U/\sqrt{\Delta V}\), when \(\Delta V>0\).